The world's unsolved problems
993 open questions across 14 fields, gathered from Wikipedia. Pick one and start thinking.
Showing all 993 problems · surprise me
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Arthur's conjectures
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Dade's conjecture relating the numbers of characters of blocks of a finite group to the numbers of characters of blocks of local subgroups.
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Demazure conjecture on representations of algebraic groups over the integers.
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Kazhdan–Lusztig conjectures relating the values of the Kazhdan–Lusztig polynomials at 1 with representations of complex semisimple Lie…
Kazhdan–Lusztig conjectures relating the values of the Kazhdan–Lusztig polynomials at 1 with representations of complex semisimple Lie groups and Lie algebras.
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McKay conjecture
in a group G, the number of irreducible complex characters of degree not divisible by a prime number p is equal to the number of irreducible complex characters of the normalizer of any Sylow p-subgroup within G.
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The Brennan conjecture
estimating the integral powers of the moduli of the derivative of conformal maps into the open unit disk, on certain subsets of \mathbb{C}
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Fuglede's conjecture on whether nonconvex sets in \mathbb{R} and \mathbb{R}^{2} are spectral if and only if they tile by translation.
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Goodman's conjecture on the coefficients of multivalued functions
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Invariant subspace problem
does every bounded operator on a complex Banach space send some non-trivial closed subspace to itself?
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Kung–Traub conjecture on the optimal order of a multipoint iteration without memory
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Lehmer's conjecture on the Mahler measure of non-cyclotomic polynomials
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The mean value problem
given a complex polynomial f of degree d \ge 2 and a complex number z, is there a critical point c of f such that |f(z)-f(c)| \le |f'(z)||z-c|?
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The Pompeiu problem on the topology of domains for which some nonzero function has integrals that vanish over every congruent copy
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Vitushkin's conjecture on compact subsets of \mathbb{C} with analytic capacity 0
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What is the exact value of Landau's constants, including Bloch's constant?
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Regularity of solutions of Euler equations
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Convergence of Flint Hills series
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Regularity of solutions of Vlasov–Maxwell equations
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Are there infinitely many Lehmer pairs?
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The 1/3–2/3 conjecture
does every finite partially ordered set that is not totally ordered contain two elements x and y such that the probability that x appears before y in a random linear extension is between 1/3 and 2/3?
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The Dittert conjecture concerning the maximum achieved by a particular function of matrices with real, nonnegative entries satisfying a…
The Dittert conjecture concerning the maximum achieved by a particular function of matrices with real, nonnegative entries satisfying a summation condition
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Problems in Latin squares
open questions concerning Latin squares
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The lonely runner conjecture
if k runners with pairwise distinct speeds run round a track of unit length, will every runner be "lonely" (that is, be at least a distance 1/k from each other runner) at some time?
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Superpermutations
smallest possible string of n digits that contains all possible permutations of n
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Map folding
various problems in map folding and stamp folding.
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No-three-in-line problem
how many points can be placed in the n \times n grid so that no three of them lie on a line?
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Rudin's conjecture on the number of squares in finite arithmetic progressions
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The sunflower conjecture
can the number of k size sets required for the existence of a sunflower of r sets be bounded by an exponential function in k for every fixed r>2?
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Frankl's union-closed sets conjecture
for any family of sets closed under sums there exists an element (of the underlying space) belonging to half or more of the sets
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Give a combinatorial interpretation of the Kronecker coefficients
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Data last refreshed 2026-10-20 03:01 UTC.